HW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
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1 CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall
2 Today's learning goals Translate sentences from English to propositional logic using appropriate propositional variables and boolean operators. Truth tables: negation, conjunction, disjunction, exclusive or, conditional, biconditional operators. Evaluate the truth value of a compound proposition given truth values of its constituent variables. Form the converse, contrapositive, and inverse of a given conditional statement. Decide and justify whether or not a collection of propositions is consistent.
3 Logic Rosen Section 1.1 Use gates and circuits to express arithmetic. Precisely express theorems and invariant statements. Make valid arguments to prove theorems.
4 Circuits Propositions 0 (off) False 1 (on) (Order switched!) True p q p v q p ^ q p q T T T T F T F T F T F T T F T F F F F F
5 Definitions Rosen p. 2-4 Proposition: declarative sentence that is T or F (not both) Propositional variable: variables that represent propositions. Compound proposition: new propositions formed from existing propositions using logical operators. Truth table: table with 1 row for each of the possible combinations of truth values of the input and an additional column that shows the truth value of the result of the operation corresponding to a particular row.
6 Truth table: (p v q) v (p v r) How many rows are in the truth table for (p v q) v (p v r)? A. 1 B. 2 C. 3 D. 4 E. None of the above How do we fill in these rows? - Inputs - Output Precedence order of operations on p. 11
7 Truth tables Can use truth table to compute truth value of compound proposition. Also, can specify logical operator by truth table.
8 Truth tables Can use truth table to compute value of compound proposition. p q r (p v q) v (p v r) T T T T T F T F T T F F F T T F T F F F T F F F
9 Compound propositions Rosen p. 3-4, 21 p T F p F T Negation p q p v q p OR q p ^ q p AND q p q p XOR q T T T T F T F T F T F T T F T F F F F F Disjunction Conjunction p p p q p q p q p q Inverter OR gate AND gate XOR gate
10 Compound propositions Rosen p. 10 Consider the compound proposition p q T T? T F? F T? F F? To fill in rows Plug in values one row at a time. OR Use intermediate columns.
11 Compound propositions Rosen p. 10 Consider the compound proposition p q T T? T F? F T? F F?
12 Compound propositions Rosen p. 10 Consider the compound proposition p q T T F F F T T F F T T F F T T F T F F F T T T F Does this look familiar?
13 Logical equivalences Rosen p. 25 Compound propositions that have the same truth values in all possible cases are logically equivalent, denoted. p q T T T T F F F T F F F F What compound proposition is logically equivalent to? A. B. C. D. E. None of the above.
14 Tautology and contradiction Rosen p. 25 Tautology: compound proposition that is always T Contradiction: compound proposition that is always F p q F T T T F T T F F T F T F T F F F T Which of the following is a tautology? A. B. C. D. E.
15 Tautology and contradiction Rosen p. 25 Tautology: compound proposition that is always T Contradiction: compound proposition that is always F p q F T T T F T T F F T F T F T F F F T Are all compound propositions either tautologies or contradictions?
16 Translation Rosen p. 22: 1.2#7 Express the sentence "The message was sent from an unknown system but it was not scanned for viruses" using the propositions p: "The message is scanned for viruses" q: "The message was sent from an unknown system" A. B. C. D. E. None of the above.
17 System specification + consistency Rosen p. 18 Can such a system be built? System specifications are consistent if they do not contain conflicting requirements In other words: the specifications are consistent if there is a truth assignment to the input propositional variables that makes each specification true.
18 System specification + consistency Rosen p. 18 System specifications are consistent if they do not contain conflicting requirements Practically speaking Start with system specifications in English Translate to compound propositions Fill in truth table with one column for each of the specifications Look for row in truth table where each output column evaluates to T
19 Conditionals Rosen p p q p à q T T T T F F F T T F F T "If p, then q"
20 Conditionals Rosen p p q p à q T T T T F F F T T F F T "If p, then q" The only way to make a conditional statement false is to
21 Conditionals Rosen p Hypothesis Antecedent Conclusion Consequent p q p à q T T T T F F F T T F F T "If p, then q"
22 Conditionals Rosen p p q p à q q à p T T T T F F F T T F F T Converse of pà q Contrapositive of pà q Inverse of pà q
23 Conditionals Rosen p p q p à q q à p T T T T F F F F T T F F F F T Converse of pà q Contrapositive of pà q Inverse of pà q
24 Conditionals Rosen p p q p à q q à p T T T T T T T F F T F T F T T F T F F F T T T T Converse of pà q Contrapositive of pà q Inverse of pà q
25 Biconditionals Rosen p Which of these compound propositions is logically equivalent to? A. B. C. D. E. None of the above. "If and only if" "Necessary and sufficient" p q T T T T F F F T F F F T
26 Biconditionals Rosen p "If and only if" "Necessary and sufficient" Notice: Compound propositions A and B are logically equivalent iff Aß à B is a tautology p q T T T T F F F T F F F T
27 Translation Rosen p. 22: 1.2#7 Express the sentence "The message is scanned for viruses whenever the message was sent from an unknown system" using the propositions p: "The message is scanned for viruses" q: "The message was sent from an unknown system" A. B. C. D. E. None of the above.
28 System specification + consistency Rosen p. 23 #11 The router can send packets to the edge system only if it supports the new address space. For the router to support the new address space, it is necessary that the latest software release be installed. The router can send packets to the edge system if the latest software release is installed. The router supports the new address space.
29 System specification + consistency Rosen p. 23 #11 p only if q. For q, it is necessary that r. p if r. q.
30 System specification + consistency Rosen p. 23 #11 p only if q. For q, it is necessary that r. p if r. q.
31 System specification + consistency Rosen p. 18 p only if q. For q, it is necessary that r. p if r. Can such a system be built? q. System specifications are consistent if they do not contain conflicting requirements In other words: the specifications are consistent if there is a truth assignment to the input propositional variables that makes each specification true.
32 System specification + consistency Rosen p. 23 #11 p q r T T T T T T T T T F T F T T T F T F T T F T F F F T T F F T T T T F T F T F T F T T F F T T T F F F F F T T T F
33 System specification + consistency Rosen p. 18 p q r T T T T T T T T T F T F T T T F T F T T F T F F F T T F F T T T T F T F T F T F T T F F T T T F F F F F T T T F System specifications are consistent if they do not contain conflicting requirements
34 Reminders Discussion section tomorrow Review quiz "due" tomorrow HW2 due Friday
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